Exponent equation (surds and indices): Given {(2^4)^(1/2)}^? = 256, determine the value of the unknown exponent ?.

Introduction / Context:
This problem assesses manipulation of exponents and the power-of-a-power rule. We are given a nested exponent expression involving powers of 2 and asked to find the unknown exponent that makes the overall value 256.

Given Data / Assumptions:

• (2^4)^(1/2) is raised to an unknown power ?.
• The resulting value equals 256.
• All operations are with real numbers and standard index laws apply.

Concept / Approach:
Use the rules: (a^m)^n = a^(m*n) and a^(m)*a^(n) = a^(m+n). Also recognize common powers of 2: 2^8 = 256. Reduce the inner expression first, then match the target value by equating exponents.

Step-by-Step Solution:
(2^4)^(1/2) = 2^(4*1/2) = 2^2 = 4So {(2^4)^(1/2)}^? = 4^?We need 4^? = 256.Note 4 = 2^2 and 256 = 2^8.(2^2)^? = 2^(2?) = 2^8 ⇒ 2? = 8 ⇒ ? = 4

Verification / Alternative check:
Compute directly: 4^4 = 16*16 = 256, confirming the result.

Why Other Options Are Wrong:
1 and 2 yield 4 and 16 respectively, not 256; 8 and 16 overshoot drastically: 4^8 and 4^16 are much larger than 256.

Common Pitfalls:
Confusing (2^4)^(1/2) with 2^(4/2) is fine, but do not treat it as 2^(1/2)*4. Always apply the power-of-a-power rule correctly.

Final Answer:
4